One obvious means we have for determining the numerosity of a set is to purposefully count the members of a set, to apply our list of number words to items in the set. Psychologists Rochel Gelman and Randy Gallistel, in their classic book The Child's Understanding of Number, outlined the functional principles required of any and all counting procedures. The Stable-Order Principle specifies that the number labels must be used in a consistent order across all counts. The One-to-One Correspondence Principle specifies that one and only one number label must be applied to each item to be counted, and the Cardinality Principle specifies that the number label applied to the final item in a count serves to represent the total number, or cardinality, of the set of items counted. Furthermore, the Order-Irrelevance Principle states that the same answer will be obtained regardless of the order in which the items in the set are counted. Finally, the Abstraction Principle states that any items can be
Encyclopedia of the Human Brain Volume 3
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counted; items need not be homogeneous to be grouped into a single count.
To engage in this procedure clearly requires having learned both the counting routine and the number words of one's language. But there is evidence that our ability to determine number is not entirely language-dependent. Human adults possess other cognitive processes that will also yield numerical representations.
B. Subitization and Estimation
It has been known for a long time that human adults can identify small numbers of objects precisely without having to consciously count them. If someone places a small handful of up to three or four coins onto a table, one would have the experience of immediately being able to tell how many there were, without counting. This ability is known as subitization and has been studied experimentally since the 1940s. There are two experimental findings that typify the subitization phenomenon: (1) If varying numbers of items are visually presented very briefly (say, 200 msec) to adult subjects whose task is to indicate the number of the items, subjects tend to show virtually perfect performance for numbers up to three or four items, with performance error increasing linearly with numbers beyond this range. (2) If visual displays containing varying numbers of items are presented to subjects whose task is to respond as accurately as possible with the number of items, subjects show very little increase in reaction time as the number of items in the display increases from zero to three or four, and then show a much steeper linear increase with additional items (see Fig. 1). Note that the first measure in essence holds reaction time constant (by presenting all stimuli equally briefly) and measures the effect of number on accuracy, whereas the second measure holds accuracy constant (by asking subjects to be as accurate as possible) and measures the effect of number on reaction time. Thus, there is a trade-off between accuracy and reaction time for numbers above the subitization range, typically at about four items and higher, but little trade-off between the two for numbers within the subitizing range.
Adults' ability to determine number without conscious counting is not limited to small set sizes. We can
Figure 1 Typical reaction-time function for subitization.
123456789 10 Number of items
Figure 1 Typical reaction-time function for subitization.
estimate the number of items in a display containing many elements, for example, the number of black dots on a page filled with dots. In such situations, whereas the exact number estimated may be quite off the actual number, adults quite reliably give much higher estimates when presented with, say, 2000 dots than when presented with, say, 500 dots. Moreover, adults are good at representing both the absolute and relative frequencies of many different kinds of items: visual entities, sounds, events, number of words in a list that begin with the letter r, and so on. This is true even when subjects have no knowledge, when presented with the stimuli, that the task will involve subsequently giving a numerical judgment.
3. Theories of the Cognitive Processes Underlying Subitization and Estimation
What is the nature of each of these abilities to determine and represent number? There have been different proposals as to the underlying nature of the subitization process. One early proposal was that subitization is a visual pattern-recognition process, in which subjects are not actually determining the number of items present in a display but are recognizing distinct patterns that typify displays of a given number. For example, two points define a line and three points in random configuration typically define a triangle. Subjects could very quickly recognize such patterns and simply associate each pattern with its corresponding numerosity. This would explain the relatively flat reaction time slope for numbers in the one to three range; the different patterns (point, line, triangle) associated with these numbers are all recognizable in about the same amount of time. However, as the number of items increases to four or more, the number of possible configurations for a given number increases exponentially, and the variety of configurations for displays of different numbers begin to overlap each other much more heavily, so that pattern recognition would break down at about this number, precisely where the subitization reaction-time data show a sharp "elbow." In this proposal, subitization (which ends at about three items) and estimation (which begins at around four and more items) are inherently distinct processes. However, this theory has largely fallen out of favor, primarily because data show that subitization effects are obtained even when subjects could not be identifying each number with a unique perceptual pattern (for example, when all points are arranged collinearly or when objects are not simple dots but complex household items each with a unique contour, making the overall contour of even a one- or two-item display complex and variable rather than linear).
A very different possibility is that subitization is a rapid and automatic serial enumeration process: subjects are determining the number of items per se in the displays through some form of unconscious counting. One specific model of such a counting process is the accumulator mechanism. This model will be described in greater detail later. Briefly, in the accumulator model, numerosities are represented by magnitudes, so that the counting process is akin to filling a bucket with a burst of water, one burst for each item to be counted. The final fullness level of the bucket represents the total number of the items counted. Evidence favoring the view that subitization is some form of serial counting process is that the reaction time curve is not flat for the numbers one to three but shows a shallow slope even across this range indicating some "cost" to each additional item in terms of processing time—a hallmark of serial processing.
How does this theory account for the subitization "elbow" in reaction times and for the decrease in accuracy with larger numbers? In this theory, there is variability in the counting process, an error term that is additive with each item counted (the bursts of water are not all identical in amount). Thus, with larger counts there will be more variance in the final fullness level. This means that larger numbers will be more difficult to discriminate than smaller numbers, leading to a decrease in accuracy as number increases. Whereas the mean fullness levels for, say, five and six will be as far apart as the mean fullness levels for two and three, the normal distributions of the fullness levels for the former will overlap more than those for the latter. In order for subjects to maintain near-perfect accuracy in subitization tasks for numbers large enough that their fullness values are not reliably discriminable, subjects will have to resort to other measures, such as consciously counting the items. When consciously counting, one must serially attend to each item in the display, such that there is a constant increase in processing time required for each added item to be counted, thus explaining the linear increase in reaction time (RT) for numbers four and higher. In this proposal, both subitization and estimation reflect the same enumeration process at work: because of the additive nature of the variance, this automatic enumeration process enables highly reliable discrimination and fast reaction time for very small numbers of items (i.e., subitization) but enables only approximate, "ballpark" identification of larger numbers (estimation).
Further evidence suggests that there may be additional processes involved in subitization as well. Lana Trick and Zenon Pylyshyn have suggested that sub-itization may be based in part on certain encapsulated processes within visual cognition. Specifically, they propose that the visual system comes equipped with a small number (three or four) of pointers or "fingers," which pick out selected objects in the visual field. Each pointer is assigned to an object the visual system wishes to track and indexes the current location of the object, updating the location as required (as the object moves, for example, or over brief occlusions and successive saccades). These fingers are assigned in parallel during the early preattentive stages of visual processing. In order to determine the numerosity of a group of items, one must first individuate the items, keeping track of the items to be counted and mentally keeping separate the already-counted items from the to-be-counted items. After individuating each item, one must operate over each item serially, for example, assigning a number word (if one is engaged in verbal counting) for each item or incrementing an accumulator with one increment for each item (the accumulator model). If there are specialized processes within the visual field for rapidly and in parallel individuating three to four items in a display, then this will enable the serial enumeration operation to commence significantly earlier, considerably speeding up—and rendering more error-free—the process for displays containing at most three or four items. In displays containing larger numbers of items, some of the items will be individuated "for free'' by these preattentive pointers within the visual system, but the remainder of the items will need to be individuated serially themselves, lengthening the time course of the process and also increasing the likelihood of error.
Evidence for such a component within the visual system comes from numerous sources. First, when subjects are presented with displays in which serial attentional processes are required to individuate the items to be enumerated (for example, displays containing items with overlapping contours such as concentric circles or containing distractor items from which the target items do not "pop out'' but must be searched for serially), the subitization function disappears: there is a linear increase in reaction time for each added number of items, even in going from one to two items. Moreover, neuropsychological data obtained by Stanislas Dehaene and Laurent Cohen showed a clear dissociation between enumeration within the subitiz-ing range and enumeration beyond this range. Their subjects were brain-lesioned human adults who were severely simultagnosic, that is, who showed significant deficits in serial visual exploration but whose parallel preattentive visual processes were largely intact. By hypothesis, these patients should not be hindered in subitizing, if subitizing is dependent on parallel, preattentive visual processes. However, they should show severe deficits in their ability to engage in serial counting processes, those hypothesized to be required for numerosities of four or more. All patients showed an intact ability to accurately quantify smaller numbers, one, two, and sometimes three, but showed significantly impaired ability to enumerate larger sets of items, showing that subitization can be preserved even when counting is impaired.
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